A recent study by Ishii and Tromp (1999) has revived
a long-lasting controversy regarding
whether or not lateral variations in density in the mantle can be constrained
using normal mode data. In principle, low angular degree normal modes are sensitive to 3D structure in all
three elastic parameters: , , and density (). However, the sensitivity to is
significantly smaller than that to or , making it difficult to
resolve even the 1D mantle density profile, and casting doubts as to
the feasibility of resolving in 3D.

Using a collection of normal mode splitting data to test inversions for 3D elastic structure up
to degree 8, Resovsky and Ritzwoller (1999)
showed that the resulting density distribution depends strongly on a priori
constraints on the model parametrization and regularization.
Recently, Kuo and Romanowicz (2000) inverted normal mode spectral
waveforms to illustrate how, depending on the parametrization and the starting
models in and , significantly different final density models can be obtained. In the latter
study, lateral structure up to spherical harmonics degree 6 was considered.

A prominent feature of the Ishii and Tromp (1999) model is that
high density regions correspond to low seismic velocities, in the central
Pacific and under Africa, in the bottom 200-500 km of the mantle. If robust,
this structure bears important implications for the dynamics and mineral
physics of the mantle. This feature has a strong degree
2 component. Since higher order normal mode splitting coefficients are still
subject to large measurement uncertainties, whereas different authors
agree on the values of most degree 2 coefficients, we reanalyze the degree 2 data to try and further
clarify the issue of resolution of 3D density structure in the mantle.
This analysis also allows us to confirm the behavior,
at the longest wavelengths, of the ratio
, which appears to increase with depth in the lower mantle
(e.g. Robertson and Woodhouse, 1995; Su and Dziewonski, 1997).

Degree 2 splitting coefficients were measured recently
by various authors. We only consider mantle modes
with no sensitivity in the inner core.
We exclude modes for which measurements
differ significantly between authors. In particular, only 13 toroidal
modes are kept, for which at least two compatible measurements exist,
or for which the unique measurement agrees with the
predictions of the SH tomographic model SAW12D (Li and Romanowicz, 1996).
Several layered parametrizations are considered. Data are
corrected for crustal structure using an isostatically compensated Moho model based on Etopo5 topography and bathymetry. The details of the crustal corrections have no
incidence on the lower mantle structure retrieved. Since
the resolution of this low angular order splitting dataset is poor in
the upper mantle, we focus the discussion on the lower mantle results.

Figure 29.1:
Maps of degree 2 in , and at representatives depths in the
lower mantle, for a model obtained by inverting the 3 parameters independently.
Note the low densities in the central Pacific at a depth of 2800km.

In our inversions, we consider overall norm damping parameters that can be adjusted
separately for , and as well as for topography of the 670-km
discontinuity (d670) and the core-mantle boundary (CMB). No regularization scheme is
applied to better assess where instabilities arise in the models. The damping parameters are adjusted so that (1) on average,
the amplitudes of the depth profiles of individual degree 2 coefficients in match
those of recent S tomographic models; (2) =
matches the range of 1.5 to 2 predicted by
mineral physics and obtained in previous studies in the top 1500km of the mantle;
(3) when inverting independently for density, =
is on average between 0.2 and 0.3 in the mid-mantle, compatible with predictions from geodynamics and mineral physics studies
(e.g. Forte and Woodward, 1997;
Karato and Karki, 2000); (4) the C20 component of the CMB topography has a value comparable with
that inferred from astronomical observations. Assigning the same damping parameter to and obtains (2) without further
adjustments, which is an indication that can be resolved independently of
. However, to obtain (3), needs to be damped at least twice as much,
and the resulting profile of is much less regular.
Figure 29.1 shows an example of a model obtained by inverting independently
for , and .

Our results show that degree 2 and structure is independently well resolved. The ratio increases significantly
below 2000km depth, confirming earlier results. Additional tests, with a parameter search on and fixed , confirm this trend, although the variance reduction achieved in such experiments is not as good as in the experiments in which and are allowed to vary independently, indicating that the assumption of perfect correlation of , and is too strong.

In a study based on body wave travel times, Bolton (1997) observed that a particular region in the Pacific
Ocean was primarily responsible for the anomalously large in the lowermost mantle. However, the global coverage was rather uneven. The fact that we observe this in degree 2 indicates that there is a global,
large scale component of heterogeneity in the lowermost mantle that cannot be
explained by thermal effects alone.

Structure in , even at degree 2, is not well resolved.
When is inverted for independently of and , the sign of in the bottom 500km of the mantle trades off with topography on the CMB. Density models based on geodynamics and mineral physics inferences are compatible with the data, whereas models with amplitudes of density heterogeneity exceeding
the latter by a factor of 2 or more can be ruled out (Figure 29.2). The mode splitting data alone cannot
resolve the existence of high density "blobs"
in the central Pacific and under Africa: models with positive correlation between and in the lowermost mantle yield slightly, but not significantly, better fits to the data (Figure 29.1), but small negative in the lowermost mantle cannot be ruled out.

The most robust feature of the degree 2 in is the increase in
at the top of the lower mantle (Figure 29.2), reaching a maximum in the depth
range 1000-1500km. Below that depth 0.3, but its sign is not
well constrained.

Figure 29.2:
From left to right: , correlations between and , and
, as a function of depth, for the 20 best (top) and 20 worst (bottom)
models obtained by inverting for and independently, and assigning
separate in different depth ranges. In all cases, the layer
parametrization is the same (from Romanowicz, 2001a).